Understanding CI for Means

1
Understanding CI for Means

• Ayona Chatterjee
• Math 2063
• University of West Georgia

2
Why do we need CI?

• We want to estimate the mean of the population,
  since we cannot measure the whole population, we
  use sample mean to estimate the population mean.
• Assume we have a small city with 20 houses and
  this is our whole population. We want to find an
  estimate for the average house price in this city.

3
House Prices
The house prices for all the houses in the
population are given in thousands of
dollars. Here the population mean is 305.1 In
real life the population size is so large that we
cannot find the true mean. There are about 90
million homes in USA. In 2007 the average house
price in USA was 299700.
House prices House prices
180 275
300 395
400 210
450 285
300 280
260 320
180 400
250 450
267 355
250 295
4

• Suppose we can only sample 10 houses from the
  population to find an estimate for the population
  mean.
• Sample 1 mean 301

House prices House prices
180 275
300 395
400 210
450 285
300 280
260 320
180 400
250 450
267 355
250 295
5

• Suppose we can only sample 8 houses from the
  population to find an estimate for the population
  mean.
• Sample 1 mean 301
• Sample 2 mean 337.5

House prices House prices
180 275
300 395
400 210
450 285
300 280
260 320
180 400
250 450
267 355
250 295
6

• Suppose we can only sample 8 houses from the
  population to find an estimate for the population
  mean.
• Sample 1 mean 301
• Sample 2 mean 337.5
• Sample 3 mean 284.2

House prices House prices
180 275
300 395
400 210
450 285
300 280
260 320
180 400
250 450
267 355
250 295
7

• Suppose we can only sample 8 houses from the
  population to find an estimate for the population
  mean.
• Sample 1 mean 301
• Sample 2 mean 337.5
• Sample 3 mean 284.2
• Sample 4 mean 300.2

House prices House prices
180 275
300 395
400 210
450 285
300 280
260 320
180 400
250 450
267 355
250 295
8

• Suppose we can only sample 8 houses from the
  population to find an estimate for the population
  mean.
• Sample 1 mean 301
• Sample 2 mean 337.5
• Sample 3 mean 284.2
• Sample 4 mean 300.2
• Sample 5 mean 342.7

House prices House prices
180 275
300 395
400 210
450 285
300 280
260 320
180 400
250 450
267 355
250 295
9
Note

• We have now five DIFFERENT estimates for the same
  population mean.
• Are all of the correct?
• How large or how small can the sample estimate
  get?
• It would be helpful to have an interval such that
  we can conclude that the population mean will lie
  within this interval with some confidence.

10
Confidence Interval

• A confidence interval is a range of values used
  to estimate the true value of a population
  parameter.
• A CI estimate of a parameter consists of an
  interval of numbers generated by a point
  estimate, together with an associated confidence
  level.

11
Confidence Level

• The confidence level is the probability 1-a that
  is the proportion of times the CI will actually
  contain the population parameter.
• Most common confidence levels are 90, 95 and
  99.
• A 95 confidence interval means that in the long
  run, the proportion of intervals that will
  contain the parameter will equal 95.